Download REFLECTION POSITIVITY, RANK

REFLECTION POSITIVITY, RANK CONNECTIVITY, AND
HOMOMORPHISM OF GRAPHS
MICHAEL FREEDMAN, LÁSZLÓ LOVÁSZ, AND ALEXANDER SCHRIJVER
Abstract. It is shown that a graph parameter can be realized as the number of homomorphisms into a fixed (weighted) graph if and only if it satisfies
two linear algebraic conditions: reflection positivity and exponential rankconnectivity. In terms of statistical physics, this can be viewed as a characterization of partition functions of vertex coloring models.
1. Introduction
For two finite graphs G and H, let hom(G, H) denote the number of homomorphisms (adjacency-preserving mappings) from G to H. Many interesting graph
parameters can be expressed in terms of the number of homomorphisms into a fixed
graph: for example, the number of colorings with a number of colors is the number
of homomorphisms into a complete graph. Further examples for the occurrence of
these numbers in graph theory will be discussed in Section 3.
Another source of important examples is statistical physics, where partition functions of various models can be expressed as graph homomorphism functions. For
example, let G be an n×n grid, and suppose that every node of G (every “site”) can
be in one of two states, “UP” or “DOWN”. The properties of the system are such
that no two adjacent sites can be “UP”. A “configuration” is a valid assignment of
states to each node. The number of configurations is the number of independent
sets of nodes in G, which in turn can be expressed as the number of homomorphisms
of G into the graph H consisting of two nodes, ”UP” and ”DOWN”, connected by
an edge, and with an additional loop at ”DOWN”. To capture more interesting
physical models, so called “vertex coloring models”, one needs to extend the notion
of graph homomorphism to the case when the nodes and edges of H have weights
(see Section 2.1).
Which graph parameters can be represented as homomorphism functions into
weighted graphs? This question is motivated, among others, by the problem of
physical realizability of certain graph parameters. Two necessary conditions are
easy to prove:
(a) The interaction between two parts of a graph separated by k nodes is bounded
by a simple exponential function of k (this will be formalized as rank-connectivity
in Section 2.2).
(b) Another necessary condition, which comes from statistical mechanics, as well
as from extremal graph theory, is reflection positivity. Informally, this means that
2000 Mathematics Subject Classification. Primary 05C99, Secondary 82B99.
Key words and phrases. graph homomorphism, partition function, connection matrix.
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M. FREEDMAN, L. LOVÁSZ, AND A. SCHRIJVER
if a system has a 2-fold symmetry, then its partition function is positive. We’ll
formulate a version of this in section 2.2.
The main result of this paper is to prove that these two necessary conditions are
also sufficient (Theorem 2.4). The proof makes use of a simple kind of (commutative, finite dimensional) C ∗ -algebras.
This result is related to characterizations of semidefinite functions on commutative semigroups. Let S be a commutative semigroup with identity. A character on
S is a real valued multiplicative function; one is interested in describing the convex
hull of characters. Let f : S → R be a function on S that is in the convex hull
of characters, then the (infinite) S × S matrix Mf defined by (Mf )s,t = f (st) is
semidefinite. Under various additional conditions on f , this necessary condition is
also sufficient; see [1, 2, 3, 6]. Among others, if Mf is positive semidefinite and has
finite rank, then f is a convex combination of a finite number of characters.
Our main result can be thought of as treating a specific semigroup (the semigroup
PLG of partially labeled graphs), and functions f on it that are invariant under
graph isomorphism (see Section 4.1); on the other hand, instead of a rank condition
on the whole matrix Mf , we consider a weaker condition bounding the rank of
certain submatrices only. Accordingly, we need infinitely many characters, but
these characters will have a finite explicit description as homomorphism numbers.
2. Homomorphisms, rank-connectivity and reflection positivity
2.1. Weighted graph homomorphisms. A weighted graph H is a graph with a
positive real weight αH (i) associated with each node i and a real weight βH (i, j)
associated with each edge ij.
Let G be an unweighted graph (possibly with multiple edges, but no loops) and
H, a weighted graph. To every homomorphism φ : V (G) → V (H), we assign the
weights
Y
αH (φ(u))
αφ =
u∈V (G)
and
(1)
homφ (G, H) =
Y
βH (φ(u), φ(v)),
uv∈E(G)
and define
(2)
hom(G, H) =
X
αφ homφ (G, H).
φ: V (G)→V (H)
homomorphism
If all the node-weights and edge-weights in H are 1, then this is the number of
homomorphisms from G into H (with no weights).
For the purpose of this paper, it will be convenient to assume that H is a complete
graph with a loop at all nodes (the missing edges can be added with weight 0). Then
the weighted graph H is completely described by a positive integer d = |V (H)|,
the positive real vector a = (α1 , . . . , αd ) ∈ Rd and the real symmetric matrix
B = (βij ) ∈ Rd×d . The graph parameter hom(., H) will be denoted by fH .
REFLECTION POSITIVITY, RANK CONNECTIVITY, AND HOMOMORPHISM OF GRAPHS 3
2.2. Connection matrices of a graph parameter. A graph parameter is a function on finite graphs (invariant under graph isomorphism). We allow multiple edges
in our graphs, but no loops. A graph parameter f is called multiplicative, if for the
disjoint union G1 ∪G2 of two graphs G1 and G2 , we have f (G1 ∪G2 ) = f (G1 )f (G2 ).
A k-labeled graph (k ≥ 0) is a finite graph in which k nodes are labeled by
1, 2, . . . k (the graph can have any number of unlabeled nodes). Two k-labeled
graphs are isomorphic, if there is a label-preserving isomorphism between them.
We denote by Kk the k-labeled complete graph on k nodes, and by Ok , the klabeled graph on k-nodes with no edges. In particular, K0 = O0 is the graph with
no nodes and no edges.
Let G1 and G2 be two k-labeled graphs. Their product G1 G2 is defined as follows:
we take their disjoint union, and then identify nodes with the same label. Hence
for two 0-labeled graphs, G1 G2 = G1 ∪ G2 (disjoint union).
Now we come to the construction that is crucial for our treatment. Let f be
any graph parameter. For every integer k ≥ 0, we define the following (infinite)
matrix M (f, k). The rows and columns are indexed by isomorphism types of klabeled graphs. The entry in the intersection of the row corresponding to G1 and
the column corresponding to G2 is f (G1 G2 ). We call the matrices M (f, k) the
connection matrices of the graph parameter f .
We will be concerned with two properties of connection matrices, namely their
rank and positive semidefiniteness. The rank r(f, k) = rk(M (f, k)), as a function
of k, will be called the rank connectivity function of the parameter f . This may be
infinite, but for many interesting parameters it is finite, and its growth rate will be
important for us. We say that a set of k-labeled graphs spans if the corresponding
rows of M (f, k) have rank r(f, k).
We say that a graph parameter f is reflection positive, if M (f, k) is positive
semidefinite for every k. This property is closely related to the reflection positivity
property of certain statistical physical models. Indeed, it implies that for any
k-labeled graph G, f (GG) ≥ 0 (since f (GG) is a diagonal entry of a positive
semidefinite matrix). Here the second copy of G can be thought of as a “reflection”
(in the set of labeled nodes) of the first.
The condition that f (GG) ≥ 0 is weaker than the condition that M (f, k) is
positive semidefinite, but we can strengthen it as follows. A quantum graph is a
formal linear combination of a finite number of finite graphs; a k-labeled quantum
graph is a formal linear combination of a finite number of finite k-labeled graphs. Let
PN
P
X = i=1 xi Gi a k-labeled quantum graph. Then X 2 = G1 ,G2 xG1 xG2 (G1 G2 ) is
a quantum graph that can be obtained by gluing together two copies of X along
the labeled nodes. Now if we extend f linearly over quantum graphs, then
f (X 2 ) = xT M (f, k)x,
showing that the non-negativity of f over symmetric quantum graphs is equivalent
to reflection positivity as we defined it.
2.3. Simple properties of connection matrices. We give a couple of simple
facts about connection matrices of a general graph parameter.
Proposition 2.1. Let f be a graph parameter that is not identically 0. Then f
is multiplicative if and only if M (f, 0) is positive semidefinite, f (K0 ) = 1, and
r(f, 0) = 1.
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M. FREEDMAN, L. LOVÁSZ, AND A. SCHRIJVER
Proof. If f is multiplicative, then f (K0 )2 = f (K0 ), showing that f (K0 ) ∈ {0, 1}.
If f (K0 ) = 0, then the relation f (G) = f (GK0 ) = f (G)f (K0 ) implies that f (G) =
0 for every G, which was excluded. So f (K0 ) = 1. Furthermore, f (G1 G2 ) =
f (G1 )f (G2 ) for any two 0-labeled graphs G1 and G2 , which implies that M (f, 0)
has rank 1 and is positive semidefinite.
Conversely, since M (f, 0) is symmetric, the assumption that r(f, 0) = 1 implies that there is a graph parameter φ and a constant c such that f (G1 G2 ) =
cφ(G1 )φ(G2 ). Since M (f, 0) is positive semidefinite, we have c > 0, and so we
can normalize φ to make c = 1. Then f (K0 ) = f (K0 K0 ) = φ(K0 )2 , whence
φ(K0 ) ∈ {−1, 1}. We can replace φ by −φ, so we may assume that φ(K0 ) = 1.
Then f (G) = f (GK0 ) = φ(G)φ(K0 ) = φ(G) for every G, which shows that f is
multiplicative.
¤
Proposition 2.2. Let f be a multiplicative graph parameter and k, l ≥ 0. Then
r(f, k + l) ≥ r(f, k) · r(f, l).
We’ll see (see Claim 4.7 below) that in the case when f is reflection positive, the
sequence r(f, k) logconvex. We don’t know if this property holds in general.
Proof. Let us call a (k+l)-labeled graph separated, if every component of it contains
either only nodes with label at most k, or only nodes with label larger than k.
Consider the submatrix of M (f, k + l) formed by the separated rows and columns.
By multiplicativity, this submatrix is the Kronecker (tensor) product of M (f, k)
and M (f, l), so its rank is r(f, k) · r(f, l).
¤
2.4. Connection matrices of homomorphisms. Fix a weighted graph H =
(a, B). For every positive integer k, let [k] = {1, . . . , k}. For any k-labeled graph
G and mapping φ : [k] → V (H), let
X
αψ
(3)
homφ (G, H) =
homψ (G, H).
αφ
ψ: V (G)→V (H)
ψ extends φ
So
(4)
hom(G, H) =
X
αφ homφ (G, H).
φ: [k]→V (H)
Lemma 2.3. The graph parameter fH = hom(., H) is reflection positive and
r(fH , k) ≤ |V (H)|k .
Proof. For any two k-labeled graph G1 and G2 and φ : [k] → V (H),
(5)
homφ (G1 G2 , H) = homφ (G1 , H)homφ (G2 , H).
The decomposition (3) writes the matrix M (f, k) as the sum of |V (H)|k matrices,
one for each mapping φ : [k] → V (H); (5) shows that these matrices are positive
semidefinite and have rank 1.
¤
The main result of this paper is a converse to Theorem 2.3:
Theorem 2.4. Let f be a reflection positive graph parameter for which there exists
a positive integer q such that r(f, k) ≤ q k for every k ≥ 0. Then there exists a
weighted graph H with |V (H)| ≤ q such that f = fH .
REFLECTION POSITIVITY, RANK CONNECTIVITY, AND HOMOMORPHISM OF GRAPHS 5
3. Examples
We start with an example showing that exponential growth of rank connectivity
is not sufficient in itself to guarantee that a graph parameter is a homomorphism
function.
Example 3.1 (Matchings). Let Φ(G) denote the number of perfect matchings in the
graph G. It is trivial that Φ(G) is multiplicative. We claim that its node-rankconnectivity is exponentially bounded:
r(Φ, k) = 2k .
Let G be a k-labeled graph, let X ⊆ [k], and let Φ(G, X) denote the number of
matchings in G that match all the unlabeled nodes and the nodes with label in X,
but not any of the other labeled nodes. Then we have for any two k-labeled graphs
G1 , G2
X
Φ(G1 G2 ) =
Φ(G1 , X1 )Φ(G2 , X2 ).
X1 ∩X2 =∅, X1 ∪X2 =[k]
This can be read as follows: The matrix M (Φ, k) can be written as a product
N T W N , where N has infinitely many rows indexed by k-labeled graphs, but only
2k columns, indexed by subsets of [k],
NG,X = Φ(G, X),
k
k
and W is a symmetric 2 × 2 matrix, where
(
1 if X1 = [k] \ X2 ,
WX1 ,X2 =
0 otherwise.
Hence the rank of M (Φ, k) is at most 2k (it is not hard to see that in fact equality
holds).
On the other hand, let us consider K1 and K2 as 1-labeled graphs. Then the
submatrix of M (Φ, 1) indexed by K1 and K2 is
µ
¶
0 1
1 0
which is not positive semidefinite. Thus Φ(G) cannot be represented as a homomorphism function.
Example 3.2 (Chromatic polynomial). The following example uses results from [4].
Let p(G) = p(G, x) denote the chromatic polynomial of the graph G. For every fixed
x, this is a multiplicative graph parameter. To describe its rank-connectivity, we
need the following notation. For k, q ∈ Z+ , let Bkq denote the number of partitions
of a k-element set into at most q parts. So Bk = Bkk is the k-th Bell number. With
this notation,
(
Bkx if x is a nonnegative integer,
r(p, k) =
Bk
otherwise.
Note that this is always finite, but if x ∈
/ Z+ , then it grows faster than ck for every
c.
Furthermore, M (p, k) is positive semidefinite if and only if either x is a positive
integer or k ≤ x + 1. The parameter M (p, k) is reflection positive if and only if this
holds for every k, i.e., if and only if x is a nonnegative integer, in which case indeed
p(G, x) = hom(G, Kx ).
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M. FREEDMAN, L. LOVÁSZ, AND A. SCHRIJVER
The following two examples show that there are important graph parameters
that are not defined as homomorphism functions, but that can also be represented
as homomorphism functions in a nontrivial way. In fact, it is easier to check that
the conditions in Theorem 2.4 hold than to show that these graph parameters are
homomorphism functions, and they first came up as counterexample candidates.
Example 3.3 (Tutte polynomial). The Tutte polynomial is a 2-variable generalization of the chromatic polynomial. It has been observed (see e.g. [9]), that this
polynomial behaves similarly along certain hyperbolas. This is also the case from
the point of view of homomorphism functions.
We in fact consider the following version T̂ (x, y) of the Tutte polynomial (which
only differs from the usual version T (x, y) by scaling). For every graph G = (V, E)
on n nodes and real parameters x, y, let
X
T̂ (G; x, y) = (1 − x)c(E) (1 − y)n T (G; x, y) =
(1 − x)c(A) (1 − y)|A|+c(A) ,
A⊆E(G)
where for A ⊆ E, c(A) denotes the number of connected components of the graph
(V, A).
Through this scaling we loose the covariance under matroid duality; but we gain
that the following “skein relation” holds for all edges e:
(6)
T (G; x, y) = (1 − y)T (G \ e; x, y) − T (G/e; x, y).
The chromatic polynomial is T̂ (1 − q, 0). One can show that the Tutte polynomial T̂ behaves just like the chromatic polynomial. Let us assume that y 6= 1
(else T (G; x, y) = 0 for all nonempty graphs). If (x − 1)(y − 1) is a positive integer, then T̂ (G; x, y) is reflection positive and has exponential rank-connectivity;
else T̂ (G; x, y) is not reflection positive and its rank-connectivity is Bk , which is
superexponential.
Thus for the case when (x − 1)(y − 1) is a positive integer, T̂ (G; x, y) can be
represented as hom(G, H) for some H. However, except for the case of the chromatic
polynomial, and the flow polynomial to be discussed in the next example, we don’t
know an explicit construction for H.
Example 3.4 (Flows). Let us start with a simple special case. Let f (G) = 1 if G is
eulerian (i.e., all nodes have even degree), and f (G) = 0 otherwise. To represent
this function as a homomorphism function, let
µ ¶
µ
¶
1/2
1 −1
a=
,
B=
1/2
−1 1
It was noted by de la Harpe and Jones [5] that for the weighted graph H = (a, B)
we have hom(G, H) = f (G).
It follows that for this function, reflection positivity holds, and the rankconnectivity is at most 2k .
This example can be generalized quite a bit. Let Γ be a finite abelian group
and let S ⊆ Γ be such that S is closed under inversion. For any graph G, fix an
orientation of the edges. An S-flow is an assignment of an element of S to each
edge such that for each node v, the product of elements assigned to edges entering
v is the same as the product of elements assigned to the edges leaving v. Let f (G)
be the number of S-flows. This number is independent of the orientation.
REFLECTION POSITIVITY, RANK CONNECTIVITY, AND HOMOMORPHISM OF GRAPHS 7
The choice Γ = Z2 and S = Z2 \ {0} gives the special case above (incidence
function of eulerian graphs). If Γ = S = Z2 , then f (G) is the number of eulerian
subgraphs of G. Perhaps the most interesting special case is when |Γ| = t and
S = Γ \ {0}, which gives the number of nowhere zero t-flows, also known as the
flow polynomial, which can be expressed as T̂ (0, 1 − q).
Surprisingly, the parameter f (G) can be described as a homomorphism function.
Let Γ∗ be the character group of Γ. Let H be the complete directed graph (with
all loops) on Γ∗ . Let αχ := 1/|Γ| for each χ ∈ Γ∗ , and let
X
βχ,χ0 :=
χ−1 (s)χ0 (s),
s∈S
0
∗
for χ, χ ∈ Γ .
We show that
(7)
f = hom(., H).
Let n = |V (G)| and m = |Γ|. For any coloring φ : V (G) → S and node v ∈ V (G),
let
X
X
∂φ (v) =
φ(uv) −
φ(vu).
u∈V (G)
uv∈E(G)
u∈V (G)
vu∈E(G)
So φ is an S-flow if an only if ∂φ = 0. Consider the expression
X
Y X
A=
χ(∂φ (v)).
φ: E(G)→S v∈V (G) χ∈Γ∗
The summation over χ is 0 unless ∂φ (v) = 0, in which case it is m. So the product
over v ∈ V (G) is 0 unless φ is an S-flow, in which case it is mn . So A · m−n counts
S-flows.
On the other hand, we can expand the product over v ∈ V (G); each term will
correspond to a choice of a character ψv for each v, and so we get
X
X
Y
A=
ψv (∂φ (v)).
φ: E(G)→S ψ: V (G)→Γ∗ v∈V (G)
Here (using that ψv is a character)
Y
ψv (∂φ (v)) =
ψv (φ(uv))
u∈V (G)
uv∈E(G)
so we get that
A=
X
X
φ: E(G)→S ψ: V
(G)→Γ∗
Y
ψv (φ(vu))−1 ,
u∈V (G)
vu∈E(G)
Y
ψv (φ(uv))ψu (φ(uv))−1 .
uv∈E(G)
Interchanging the summation, the inner sum factors:
X
Y
ψv (φ(uv))ψu (φ(uv))−1
φ: E(G)→S uv∈E(G)
=
Y
X
ψv (s)ψu (s)−1
uv∈E(G) s∈S
=
Y
uv∈E(G)
βψu ψv = mn homψ (G, H),
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M. FREEDMAN, L. LOVÁSZ, AND A. SCHRIJVER
showing that
A = mn hom(G, H).
This proves (7).
Example 3.5 (The role of multiple edges). Let us give an example of a reflection
positive graph parameter f for which r(f, k) is finite for every k but has superexponential growth. The example also shows that we have to be careful with multiple
edges. Let, for each graph G, G0 denote the graph which we obtain from G by
keeping only one copy of each parallel class of edges. Let
0
f (G) = 2−|E(G )| .
It is not hard to see that the connection matrices M (f, k) are positive semidefinite.
This graph parameter is, in fact, the limit of parameters of the form hom(., H):
take homomorphisms into a random graph H = G(n, 1/2), with all node weights
= 1/n and all edge-weights 1. Furthermore, it is not hard to see that the rank of
k
M (f, k) is 2(2) . This is finite but superexponential, so the parameter is not of the
form hom(., H).
Note, however, that for a simple graph G (i.e., if G has no multiple edges),
f (G) = 2|E(G)| can be represented as the number of homomorphisms into the
graph consisting of a single node with a loop, where the node has weight 1 and the
loop has weight 1/2.
Example 3.6 (Homomorphisms into infinite graphs). We can extend the definition
of hom(G, H) to infinite weighted graphs H provided the node and edge-weights
form sufficiently fast convergent sequences. Reflection positivity remains valid,
but the rank of M (fH , k) will become infinite, so this graph parameter cannot be
represented by a finite H.
More generally, let a > 0, I = [0, a], and let W : I × I → R be a measurable
function such that for every n ∈ Z+ ,
Z aZ a
|W (x, y)|n dx dy < ∞.
0
0
Then we can define a graph parameter fW as follows. Let G be a finite graph on n
nodes, then
Z
Y
fW (G) =
W (xi , xj ) dx1 . . . dxn .
I n ij∈E(G)
It is easy to see that for every weighted (finite or infinite) graph H, the graph
parameter fH is a special case. Furthermore, fW is reflection positive.
However, it can be shown that the graph parameter in Example 3.5 cannot be
represented in this form. A characterization of graph parameters representable in
this more general form is given in [7].
4. Proof of Theorem 2.4
4.1. The algebra of graphs. In this first part of the proof, we only assume that
for every k, M (f, k) is positive semidefinite, has finite rank rk , and r0 = 1. We
know that f is multiplicative, and hence f (K0 ) = 1. We can replace the parameter
f (G) by f (G)/f (K1 )−|V (G)| ; this can be reversed by scaling of the node-weights of
the target graph H representing f , once we have it constructed. So we may assume
REFLECTION POSITIVITY, RANK CONNECTIVITY, AND HOMOMORPHISM OF GRAPHS 9
that f (K1 ) = 1. Combined with multiplicativity, this implies that we can delete
(or add) isolated nodes from any graph G without changing f (G).
It will be convenient to put all k-labeled graphs into a single structure as follows.
By a partially labeled graph we mean a finite graph in which some of the nodes
are labeled by distinct nonnegative integers. Two partially labeled graphs are
isomorphic if there is an isomorphism between them preserving all labels. For two
partially labeled graphs G1 and G2 , let G1 G2 denote the partially labeled graph
obtained by taking the disjoint union of G1 and G2 , and identifying nodes with the
same label. This way we obtain a commutative semigroup PLG. For every finite
set S ⊆ Z+ , we call a partially labeled graph S-labeled, if its labels form the set S.
It may be useful at this point to discuss the case we are aiming at, namely when
f = hom(., H) for some weighted graph H. Our assumption that f (K1 ) = 1 means
then that the sum of nodeweights is 1. For every map φ : Z+ → V (H), and every
partially labeled graph G with label set S, we define homφ (G, H) = homφ|S (G, H).
Equation (5) implies that the function homφ (., H) is a character of the semigroup
PLG, and formula (4) gives a representation of f as a convex combination of these
(very special) characters.
Let G denote the (infinite dimensional) vector space of formal linear combinations
(with real coefficients) of partially labeled graphs. We can turn G into an algebra by
using G1 G2 introduced above as the product of two generators, and then extending
this multiplication to the other elements linearly. (So G is the semigroup algebra
of PLG.) Clearly G is associative and commutative, and the empty graph is a unit
element.
For every finite set S ⊆ Z+ , the set of all formal linear combinations of S-labeled
graphs forms a subalgebra G(S) of G. The graph with |S| nodes labeled by S and
no edges is a unit in this algebra, which we denote by US .
We can extend f to a linear functional on G, and define an inner product
hx, yi = f (xy)
for x, y ∈ G. By our hypothesis that f is reflection positive, this inner
P product is
positive semidefinite, i.e., hx, xi ≥ 0 for all x ∈ G. Indeed, if x = G∈G xG G ∈ G
(with a finite number of nonzero terms), then
X
hx, xi = f (xx) =
xG1 xG2 f (G1 G2 ) ≥ 0,
G1 ,G2 ∈G
since the quadratic form is of the form xT M (f, k)x for a large enough k, and M (f, k)
is positive semidefinite.
Let
K = {x ∈ G : f (xy) = 0 ∀y ∈ G}
be the annihilator of G. It follows from the positive semidefiniteness of the inner
product that x ∈ K could also be characterized by f (xx) = 0.
Clearly, K is an ideal in G, so we can form the quotient algebra Ĝ = G/K. We can
also define Ĝ(S) = G(S)/K. It is easy to check that every graph US has the same
image under this factorization, namely unit element u of Ĝ. Furthermore, if x ∈ K
then f (x) = f (xu) = 0, and so f can also be considered as a linear functional on
Ĝ. We denote by Ĝ the element of Ĝ corresponding to the partially labeled graph
G.
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M. FREEDMAN, L. LOVÁSZ, AND A. SCHRIJVER
Next, note that Ĝ(S) is a finite dimensional commutative algebra of dimension
r|S| , with the positive definite inner product hx, yi = f (xy).
Let S ⊆ T be finite subsets of Z+ . Then Ĝ(S) ⊆ Ĝ(T ). Indeed, every S-labeled
graph G can be turned into a T -labeled graph G0 by adding |T \ S| new isolated
nodes, and label them by the elements of T \ S. It is straightforward to check that
G − G0 ∈ K, and so G and G0 correspond to the same element of Ĝ.
We’ll also need the orthogonal projection πS of Ĝ to the subalgebra Ĝ(S). This
has a very simple combinatorial description. For every partially labeled graph G
and S ⊆ Z+ , let GS denote the S-labeled graph obtained by deleting the labels not
in S; then πS (Ĝ) = ĜS .
To see this, let G be any partially labeled graph. Then
hGS , G − GS i = f (GS (G − GS )) = f (GS G) − f (GS GS ) = 0,
since GGS and GS GS are isomorphic graphs. Hence
hĜS , Ĝ − ĜS i = 0,
showing that ĜS is indeed the orthogonal projection of Ĝ onto Ĝ(S).
We’ll be interested in the idempotent elements of Ĝ. If p is idempotent, then
f (p) = f (pp) = hp, pi > 0.
For two idempotents p and q, we say that q resolves p, if pq = q. It is clear that
this relation is transitive.
Let S be a finite subset of Z+ , and set r = r|S| . Since the algebra Ĝ(S) is finite
dimensional and commutative, and all its elements are self-adjoint with respect
to the positive definite inner product h., .i, it has a (uniquely determined) basis
PS = {pS1 , . . . , pSr } such that (pSi )2 = pSi and pSi pSj = 0 for i 6= j. We denote
by PT,p the set of all idempotents in PT that resolve a given idempotent p. If
|T | = |S| + 1, then the number of elements in PT,p will be called the degree of
p ∈ PS , and denoted by deg(p). Obviously this value is independent of which
(|S| + 1)-element superset T of S we are considering.
Claim 4.1. Let r be any idempotent element of G(S). Then r is the sum of those
idempotents in PS that resolve it.
Indeed, we can write
r=
X
µp p
p∈PS
with some scalars µp . Now using that r is idempotent:
X
X
r = r2 =
µp µp0 pp0 =
µ2p p,
p,p0 ∈PS
p∈PS
µ2p
which shows that
= µp for every p, and so µp ∈ {0, 1}. So r is the sum of some
subset X ⊆ PS . It is clear that rp = p for p ∈ X and rp = 0 for p ∈ PS \ X, so X
consists of exactly those elements of PS that resolve q.
As a special case, we see that
X
u=
p
(8)
p∈PS
is the unit element of Ĝ(S) (this is the image of the graph US ), and also the unit
element of the whole algebra Ĝ.
REFLECTION POSITIVITY, RANK CONNECTIVITY, AND HOMOMORPHISM OF GRAPHS11
Claim 4.2. Let S ⊂ T be two finite sets. Then every q ∈ PT resolves exactly one
element of PS .
Indeed, we have by (8) that
X
u=
p=
p∈PS
and also
u=
X
X
p∈PS
q∈PT
q resolves p
X
q,
q,
q∈PT
so by the uniqueness of the representation we get that every q must resolve exactly
one p.
Claim 4.3. Let S, T, U be finite sets and S = T ∩ U . If x ∈ Ĝ(T ) and y ∈ Ĝ(U ),
then
f (xy) = f (πS (x)y).
Indeed, for every T -labeled graph G1 and U -labeled graph G2 , the graphs G1 G2
and πS (G1 )G2 are isomorphic. Hence the claim follows by linearity.
Claim 4.4. If p ∈ PS and q resolves p, then
πS (q) =
f (q)
p.
f (p)
We show that both sides give the same inner product with every basis element
in PS . Since q does not resolve any p0 ∈ PS \ {p}, we have p0 q = 0 for every such
p0 . By Claim 4.3, this implies that
hp0 , πS (q)i = f (p0 πS (q)) = f (p0 q) = 0 = hp0 ,
f (q)
pi.
f (p)
hp, πS (q)i = f (pπS (q)) = f (pq) = f (q) = hp,
f (q)
pi.
f (p)
Furthermore,
This proves the Claim.
Claim 4.5. Let S, T, U be finite sets and S = T ∩U . Then for any p ∈ PS , q ∈ PT,p
and r ∈ Ĝ(U ) we have
f (p)f (qr) = f (q)f (pr).
Indeed, by Claims 4.4 and 4.3,
f (qr) = f (πS (q)r) =
f (q)
f (pr).
f (p)
Claim 4.6. If both q ∈ PT and r ∈ PU resolve p, then qr 6= 0.
Indeed, by Claim 4.5,
f (qr) =
f (q)
f (q)
f (pr) =
f (r) > 0.
f (p)
f (p)
Claim 4.7. If S ⊂ T , and q ∈ PT resolves p ∈ PS , then deg(q) ≥ deg(p).
12
M. FREEDMAN, L. LOVÁSZ, AND A. SCHRIJVER
It suffices to show this in the case when |T | = |S| + 1. Let U ⊂ Z+ be any
(|S| + 1)-element superset P
of S different from T . Let Y be the set of elements in
PU resolving p. Then p = r∈Y r by Claim 4.1. Here |Y | = deg(p). Furthermore,
we have
X
X
rq = q
r = qp = q.
r∈Y
r∈Y
Each of the terms on the left hand side is nonzero by Claim 4.6, and since the
terms are all idempotent, each of them is the sum of one or more elements of PT ∪U .
Furthermore, if r, r0 ∈ Y (r 6= r0 ), then we have the orthogonality relation
(rq)(r0 q) = (rr0 )q = 0,
so the basic idempotents in the expansion of each term are different. So the expansion of q in PT ∪U contains at least |Y | = deg(p) terms, which we wanted to
prove.
4.2. Bounding the expansion. From now on, we assume that there is a q > 0
such that rk ≤ q k for all k.
So if a basic idempotent p ∈ PS has degree D, then there are D basic idempotents
on the next level with degree ≥ D, and hence if |T | ≥ |S|, then the dimension of
Ĝ(T ) is at least D|T \S| . It follows that the degrees of basic idempotents are bounded
by q; let D denote the maximum degree, attained by some p ∈ PS .
Let us fix such a set S and p ∈ PS with maximum degree D. For u ∈ Z+ \ S, let
u
denote the elements of PS∪{u} resolving p. Note that for u, v ∈ Z+ \ S,
q1u , . . . , qD
there is a natural isomorphism between Ĝ(S ∪ {u}) and Ĝ(S ∪ {v}) (induced by the
map that fixes S and maps u onto v), and we may choose the labeling so that qiu
corresponds to qiv under this isomorphism.
Next we describe, for a finite set T ⊃ S, all basic idempotents in PT that resolve
p. Let V = T \ S, and for every map φ : V → {1, . . . , D}, let
Y
v
(9)
qφ =
qφ(v)
.
v∈V
Note that by Claim 4.5,
(10)
f (qφ ) = f (
Y
v∈V
v
qφ(v)
)=
v
)¢
¡ Y f (qφ(v)
f (p) 6= 0,
f (p)
v∈V
and so qφ 6= 0.
Claim 4.8.
PT,p = {qφ : φ ∈ {1, . . . , D}V }.
We prove this by induction on |T \ S|. For |T \ S| = 1 the assertion is trivial.
Suppose that |T \ S| > 1. Let u ∈ T \ S, U = S ∪ {u} and W = T \ {u}; thus
U ∩ W = S. By the induction hypothesis, the basic idempotents in PW resolving
p are elements of the form qψ (ψ ∈ {1, . . . , D}V \{u} ).
Let r be one of these. By Claim 4.6, rqiu 6= 0 for any 1 ≤ i ≤ D, and clearly
resolves r. We can write rqiu as the sum of basic idempotents in PT , and it is easy
to see that these also resolve r. Furthermore, the basic idempotents occurring in
the expression of rqiu and rqju (i 6= j) are different. But r has degree D, so each
rqiu must be a basic idempotent in PT itself.
REFLECTION POSITIVITY, RANK CONNECTIVITY, AND HOMOMORPHISM OF GRAPHS13
Since the sum of the basic idempotents rqiu (r ∈ PW,p , 1 ≤ i ≤ D) is p, it follows
that these are all the elements of PT,p . This proves the Claim.
It is immediate from the definition that an idempotent qφ resolves qiv if and only
φ(v) = i. Hence it also follows that
X
qφ .
qiv =
(11)
φ: φ(v)=i
4.3. Constructing the target graph. Now we can define H as follows. Let H
be the looped complete graph on V (H) = {1, . . . , D}. We have to define the node
weights and edge weights.
Fix any u ∈ Z+ \ S. For every i ∈ V (H), let
αi =
f (qiu )
f (p)
be the weight of the node j. Clearly αi > 0.
Let u, v ∈ Z+ \S, v 6= u, and let W = S ∪{u, v}. Let Kuv denote the graph on W
which has only one edge connecting u and v, and let kuv denote the corresponding
element of Ĝ(W ). We can express pkuv as a linear combination of elements of PW,p
(since for any r ∈ PW \ PW,p one has rp = 0 and hence rpku,v = 0):
X
βij qiu qjv .
pkuv =
i,j
This defines the weight βij of the edge ij. Note βij = βji for all i, j, since pkuv =
pkvu .
We prove that this weighted graph H graph gives the right homomorphism function.
Claim 4.9. For every finite graph G, f (G) = hom(G, H).
By (11), we have for each pair u, v of distinct elements of V (G)
X
X
X
X
pkuv =
βi,j qiu qjv =
βi,j
qφ =
βφ(u),φ(v) qφ .
i,j∈V (H)
i,j∈V (H)
φ∈V (H)V
φ: φ(u)=i
φ(v)=j
Consider now any V -labeled graph G with V (G) = V , and let g be the corresponding element of Ĝ. Then
Y
Y ¡ X
¢
pg =
pkuv =
βφ(u),φ(v) qφ
uv∈E(G)
X
=
uv∈E(G) φ∈V (H)V
Y
¡
¢
βφ(u),φ(v) qφ .
φ:V →V (H) uv∈E(G)
Since p ∈ Ĝ(S), g ∈ Ĝ(V ) and S ∩ V = ∅, we have f (p)f (g) = f (pg), and so by
(10),
X ¡ Y
¢
f (p)f (g) = f (pg) =
βφ(u),φ(v) f (qφ )
=
X
φ∈V (H)V uv∈E(G)
¡
Y
φ:V →V (H) uv∈E(G)
βφ(u),φ(v)
¢¡ Y
¢
αφ(v) f (p),
v∈V (G)
The factor f (p) > 0 can be cancelled from both sides, completing the proof.
14
M. FREEDMAN, L. LOVÁSZ, AND A. SCHRIJVER
5. Extensions: Graphs with loops, directed graphs and hypergraphs
In our arguments we allowed parallel edges in G, but no loops. Indeed, the
representation theorem is false if G can have loops: it is not hard to check that the
graph parameter
f (G) = 2#loops
cannot be represented as a homomorphism function, even though its connection
matrix M (f, k) is positive semidefinite and has rank 1. To get a representation
theorem for graphs with loops, each loop e in the target graph H must have two
weights: one which is used when a non-loop edge of G is mapped onto e, and the
other, when a loop of G is mapped onto e. With this modification, the proof goes
through.
The constructions and results above are in fact more general; they extend to
directed graphs and hypergraphs. There is a common generalization to these results,
using semigroups. This will be stated and proved in a separate paper. An analogous
result for edge coloring models (with a substantially more difficult proof) was proved
by B. Szegedy [8].
Acknowledgement
We are indebted to Christian Borgs, Jennifer Chayes, Monique Laurent, Miki
Simonovits, Vera T. Sós, Balázs Szegedy, Gábor Tardos and Kati Vesztergombi for
many valuable discussions and suggestions on the topic of graph homomorphisms.
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